A lower bound for the spectral radius of a digraph

We show that the spectral radius ρ(D) of a digraph D with n vertices and c₂ closed walks of length 2 satisfies Moreover, equality occurs if and only if D is the symmetric digraph associated to a -regular graph, plus some arcs that do not belong to cycles. As an application of this result, we construct new sharp upper bounds for the low energy of a digraph, which extends Koolen and Moulton bounds of the energy to digraphs.